Canonical Quantum Gravity for Philosophers (of Physics)

Transcription

1 Canonical Quantum Gravity for Philosophers (of Physics) Seminar on the Philosophical Foundations of Quantum Gravity University of Illinois, Chicago 28 September 2013

2 1 The problem of time in canonical general relativity Hamiltonian GR The problem of time 2

3 Canonical quantum gravity Hamiltonian GR The problem of time vantage point: general relativity (GR) strategy: apply canonical quantization procedure conservative approach, attempt to stay close to old physics main families in genus: quantum geometrodynamics, loop quantum gravity

4 Background Hamiltonian GR The problem of time Huggett, Nick, Tiziana Vistarini, and CW. Time in quantum gravity. In Adrian Bardon and Heather Dyke (eds.), The Blackwell Companion to the Philosophy of Time, Chichester: Wiley-Blackwell (2013), CW. Raiders of the lost spacetime. Written for Dennis Lehmkuhl (ed.), Towards a Theory of Spacetime Theories. Boston: Birkhäuser (2014?).

5 Hamiltonian GR The problem of time Hamiltonian general relativity: pros and cons Earman, John. Tracking down gauge: an ode to the constrained Hamiltonian formalism. In K. Brading and E. Castellani (eds.), Symmetries in Physics: Philosophical Reflections. Cambridge UP (2003), Pros (many stressed by Earman 2003): explains how the fibre bundle formalism arises in the cases it does has a sufficiently broad scope to relate GR to Yang-Mills gauge theories offers a formalization of the gauge concept connects to fundamental foundational issues such as the nature of observables and the status of determinism in GR and in gauge theories natural affinity to the initial value problem in standard GR canonical quantization

6 Hamiltonian GR The problem of time Hamiltonian general relativity: pros and cons Cons: Maudlin, Tim. Thoroughly muddled McTaggart: or how to abuse gauge freedom to generate metaphysical monstrosities. Philosophers Imprint 2/4 (2002). Tim Maudlin (2002, 9): GR is forced into the Procrustean bed of the Hamiltonian formalism physical systems that are spatially extended or at least located in space, and their dynamical over time 4-dim spacetime M, g ab gets recast as 3-dim spaces which evolve in a fiducial time according to the dynamics governed by Hamilton s equation But: What Minkowski has joined together, let no man put asunder.

7 Hamiltonian GR The problem of time Hamiltonian GR: a Lagrangian basis Einstein-Hilbert action S[g ab ] for gravity without matter: S[g ab ] = 1 d 4 x gr, (1) 16πG where G is Newton s gravitational constant, g the determinant of the metric tensor g ab, and R the Ricci scalar get Lagrangian formulation of GR with Euler-Lagrange eqs replacing (vacuum) Einstein eqs if (1) is varied wrt metric g ab Euler-Lagrange eqs are second order, with solutions uniquely determined by q, q iff so-called Hessian matrix 2 L(q, q)/ q n q n of Lagrangian function L(q, q), where n labels the degrees of freedom, is invertible If not: accelerations q not uniquely determined, and solutions contain arbitrary functions of time impossibility of inverting 2 L(q, q)/ q n q n is an indication of gauge freedom M

8 Hamiltonian GR The problem of time Hamiltonian systems with constraints transpose Euler-Lagrange eqs into Hamiltonian eqs of motion, q = H/ p and ṗ = H/ q, by introducing canonical momenta via p n = L q n, (2) where n = 1,..., N (= number of DOFs), but these are not independent if Hessian is singular ( gauge freedom) dependencies get articulated in ( primary ) constraint equations φ m (q, p) = 0, m = 1,..., M, (3) where M is the number of dependencies eqs (3) define a submanifold of the phase space, the constraint surface

9 Hamiltonian GR The problem of time Hamiltonian systems with constraints (Cutting some mathematical corners...) Euler-Lagrange eqs are equivalent to union of Hamilton s equations and constraint eqs (3), and we get eq of motion for arbitrary functions F(q, p) of the canonical variables Ḟ = {F, H} + u m {F, φ m }, (4) where {, } is the usual Poisson bracket and the u m are extra variables to render the Legendre trafos invertible arbitrary functions u m in Hamiltonian eqs of motion physical state is uniquely determined by a pair (q, p), i.e., by a point in Γ, but not vice versa the u m s encode gauge freedom (CSMC...) physical phase space defined as sets of points representing gauge equivalence classes of points in Γ

10 Hamiltonian GR The problem of time Parmenides s revenge: the problem of time GR as constrained Hamiltonian system: Hamiltonian H itself is constraint bound to vanish on constraint surface problem of time, or better, problem of change crucial premise: only gauge-invariant quantities can capture genuinely physical content of theory (two distinct mathematical models of a theory describe same physical situation just in case they are related by maps which are interpreted as gauge trafos) concept of Dirac observables Definition (First-class constraints) A function F (q, p) is termed first class if and only if its Poisson bracket with every (primary and secondary) constraint vanishes weakly, {F, φ j } 0, j = 1,..., J. (5) If a first-class function is a constraint, then it s a first-class constraint.

11 Hamiltonian GR The problem of time Definition (Dirac observables) A(n equivalence class of) Dirac observable(s) is defined as the (set of those) function(s) in phase space that has (have) weakly vanishing Poisson brackets with all first-class constraints (and coincide on the constraint surface). Equivalently, Dirac observables are functions in phase space which are constant along gauge orbits on the constraint surface. If premise is true, and if the gauge-invariant quantities of a constrained Hamiltonian theory are precisely its Dirac observables as defined in Definition 2, then the physical content of a constrained Hamiltonian theory is exhausted by its Dirac observables. Go identify first-class constraints!

12 First-class constraints in HGR Hamiltonian GR The problem of time Principle (General Covariance) The Einstein equations dynamical symmetry group Diff (M) of active spacetime diffeomorphisms is the gauge group of GR. In other words, active spacetime diffeomorphisms, which map a solution of the dynamical equation to another solution, ought to be considered relating two mathematically distinct solutions describing one and the same physical situation. (Wüthrich 2006, 3.2) Hamiltonian GR: dynamical syms encoded in constraints, Diff (M) breaks down into 3-dim spatial diffeos and 1-dim temporal diffeos ADM version: 4-dim diffeos generated by normal and tangential components of Hamiltonian constraints generating diffeos, and hence these components of Hamiltonian, vanish weakly

13 The problem of change Hamiltonian GR The problem of time received interpretation of Hamiltonian: generator of dynamical evolution via Hamilton eqs Dirac observables have weakly vanishing Poisson brackets with the Hamiltonian, and are constants of the motion... There cannot be change even in the very minimal sense that the physical quantities can have different values at different times. last remnant of a dynamical aspect of time is lost, the dynamics gets frozen! any supposed change is purely a representational redundancy, and not a physical fact Question: how can time and change arise phenomenologically in a fundamentally changeless world?

14 Reactions from physics Hamiltonian GR The problem of time It really is a deep problem (apparently even underestimated by Earman!): how can a physicalist explain change that can t even supervene on the fundamental? Note: problem arises from treating Hamiltonian GR treating as we standardly treat gauge theories, i.e., from treating Diff(M) as a gauge symmetry just as gauge groups in gauge theories embrace timelessness/changelessness and explicate emergence of time/change or else identify what ought to be rejected in argument above and given an account of how to implement general covariance (Earman 2002b, S217) Examples in physics literature of the former kind include Rovelli (2011) and, somewhat more radically, Barbour (2001, 2009); Kuchař (1993) is one of the latter type. Either way, formidable challenges must be faced.

15 Pick your poison Hamiltonian GR The problem of time 1 Resistance: One way to resists conclusion would be by adding more fundamental structure with observables which change over time. Any such approach will introduce radically new interpretations of old physics or altogether new substantive physics.

16 Pick your poison Hamiltonian GR The problem of time 2 Accepters face two tasks: (1) Show how one can do physics in fundamentally non-changing world: Schrödinger eq degenerates into Wheeler-DeWitt eq Ĥ ψ = 0 in quantum mechanical description of universe, and in statistical physics, one faces difficulty of explicating how thermalization can occur in the absence of time and change, etc (2) Explain our phenomenology of temporality; serious challenge to physicalism: [i]nsofar as a physical explanation must be couched in terms of genuine observables, the sought after explanation cannot be purely physical. (Earman 2002a, 21f)

17 Reactions from philosophy (1) Richard Healey Hamiltonian GR The problem of time Healey 2002, 2004: there can be B-series change in the standard formulation of GR tries to obtain an explanation of how change can emerge from a fundamental reality which he agrees does not change Healey: same in Hamiltonian formulation, as long as we accept that the physical content is not exhausted by Dirac observables change must properly be characterized with respect to a frame of reference change cannot be captured by fundamental properties which are ex constructione frame-independent

18 Reactions from philosophy (1) Richard Healey Hamiltonian GR The problem of time philosophical counterpart of Rovelli s program of partial observables (2002, 2011), as also noticed by Rickles (2008, 337) Rovelli: construct gauge-invariant observables from correlations among partial, but gauge-dependent ones open question how Healey s and Rovelli s work relates, and whether it evades strictures of chilling conclusion

19 Reactions from philosophy (2) Tim Maudlin Hamiltonian GR The problem of time Maudlin 2002: (Bergmann) observables lack meaningful contact to what is in fact observable arising indeterminism is not physically salient, it is faux and completely phony, which he thinks can be dealt with in three ways 1 ignore it it s just an artefact of the representation doesn t work if canonical QG is taken seriously it may vindicate Ham GR follow route and let the foundational chips fall where they may

20 Reactions from philosophy (2) Tim Maudlin Hamiltonian GR The problem of time 2 fix a gauge, thus removing indeterminism may work for some physically relevant cases but: all quantifiable properties will depend on gauge chosen and hence on selected representation, not genuine physical content would need to justify gauge as physically privileged to extent to which we take symmetry to be truly gauge, fixing it and reading off deep metaphysical insights is not an option

21 Reactions from philosophy (2) Tim Maudlin Hamiltonian GR The problem of time 3 quotient out indeterminism, cut out all mathematical surplus structure by reformulating theory s.t. all previously mathematically distinct but physically identical states get represented by exactly one point in (reduced) phase space theory is freed from gauge only principled and acceptable way of dealing with the indeterminism Maudlin complains that this will lead to the rather silly, crazy, and metaphysically monstrous conclusion that there cannot be change But that s restating the problem, not solving it!

22 Hamiltonian GR The problem of time Problem of time/change: concluding remarks to extent we take Hamiltonian GR seriously, we cannot easily escape grip of problem of time/change explanatory debt remains: if borne out, how can we humans amid a fundamentally changeless universe have so vivid perceptions as of a fleeting passage of time? As long as this debt is not discharged, any approach entailing a fundamentally frozen world comes with a promissory note. To fully redeem this note is a formidable task, both technically as well as philosophically.

23 The dissolution of spacetime in QG In string theory as well as in loop quantum gravity (LQG) (and other approaches to QG), indications are coalescing that space and time are no longer fundamental entities (substantivally or relationally), but merely emergent phenomena that arise from the basic physics. Emergent here should not be understood as the terminus technicus in philosophy where an emergent property is not even weakly reducible; instead: collective designation for reductive relationships In the language of physics: spt thys such as GR as effective and spacetime itself emergent, much like thermodynamics is an effective theory and temperature is emergent, as it is built up from the collective behaviour of gas molecules. Unlike the fact that temperature is emergent, the idea that the universe is not in space and time shocks our very idea of physical existence as profoundly as any scientific revolution.

24 Loop quantum gravity starts out from Hamiltonian formulation of GR in an attempt to apply canonical quantization believed to correspond to three-dimensional spatial structure are so-called spin networks spin networks: networks of interwoven loops with spin representations on nodes and edges spin representations quantify discretely valued quantum volume (nodes) and area of adjacency surface (edges) Although dynamics of theory remains incomplete, general scheme evolves spin networks by Hamiltonian acting on them. resulting structure ( spin foam ): quantum analogue of four-dimensional spacetime

25 Quantum space in LQG: a spin network i 2 j 2 j 3 j 1 i 1

26 Quantum space in LQG: a spin network spin networks are discrete structures (no affine or differentiable structure) no natural dimensionality no metric structure, although spin networks in eigenstates of geometrically interpreted operators afford geometric interpretation space: superposition of these eigenstates, i.e. no well-defined geometric properties states in superpositions in general have different connectivity (and perhaps different cardinality) different structures, and what is local in one term of the superposition will in general not be local in others How local, i.e. topological, structures like relativistic spacetimes emerge from spin networks is at present little understood.

27 Even geometric eigenstates are non-local In LQG: locality must be explicated in terms of adjacency relations encoded in the fundamental structure. In general, two fundamentally adjacent nodes will not map in the same neighbourhood of the emerging spacetime: large distance spacetime emergence spin network

28 The problem of space in LQG I am glossing over many details here, no doubt, but it should be clear that the locality structure of spin networks captured by the combinatorial connectivity of the spin network structure is very different from the one of relativistic spacetimes although non-localities in fundamental structure are suppressed in approximation of the large scales relevant to the emergence of spacetimes. fundamental spatial structure in LQG is rather different from the one we know from GR: discrete, non-metrical, combinatorial structure with different topology and hence locality structure Real question: analogously to time, the question arises how do we recover what we know is very highly empirically successful description of space(time) from fundamental structure? We have a problem of space!

29 Are spacetime-less theories empirically incoherent? Definition (Empirical incoherence) A theory is empirically incoherent just in case the truth of the theory undermines our empirical justification for accepting it as true. (cf. Barrett 1999, 4.5.2) A theory denying the fundamental existence of spacetime is thus alleged to be empirically incoherent because the empirical justification of a theory derives only from our observation of the effects of a localized something situated in spacetime; yet if such a theory were true, there could be no such localized something. Barrett, Jeffrey A. The Quantum Mechanics of Minds and Worlds. Oxford University Press (1999).

30 Saving empirical coherence Huggett, Nick and Wüthrich, Christian. Emergent spacetime and empirical (in)coherence. Studies in the History and Philosophy of Modern Physics 44 (2013): Avoiding the charge of empirical incoherence requires that a theory must be shown to admit emergent spacetime, i.e., it must be shown how general-relativistic spacetime emerges as an approximation in quantum theories of gravity This task is also important in the context of justification, as its discharge provides an account of why the classical spacetime theory (GR) was as successful as it was.

31 Context of justification quantization: context of discovery taking the classical limit: context of (partial) justification understanding how (classical) spt re-emerges is not only important to save the appearances, accommodate common sense etc, but also a methodologically central part of the enterprise of QG (to retrieve GR as limit).

32 The present situation regarding emergence mapping from set of quantum states to set of classical spacetimes not bijective, but many-to-one and for some quantum sets, there s no classical analogue so far: classical limit of LQG has defied understanding role for philosophers: explore landscape, map possibilities, bring lit on emergence/reduction to bear on issue Bibliography of phil lit on emergence in canonical QG: Butterfield, J. and C. Isham. On the emergence of time in quantum gravity. In Butterfield (ed.), The Arguments of Time, OUP (1999), Butterfield, J. and C. Isham. Spacetime and the philosophical challenge of quantum gravity. In Callender and Huggett (ed.), Physics Meets Philosophy at the Planck Scale, Cambridge UP (2001), Wüthrich, Christian. Approaching the Planck Scale from a Generally Relativistic Point of View: A Philosophical Appraisal of Loop Quantum Gravity. PhD Thesis, University of Pittsburgh (2006).

33 The Butterfield-Isham scheme Ultimately, full analysis will depend on full articulation of theory; in the meantime, limit focus to kinematical level ( don t have to deal with problem of time, unlike Butterfield and Isham) My job: apply concepts from Butterfield and Isham for emergence of full spacetime (rather than just time) Butterfield and Isham (1999) identify three types of reductive relations between theories, the third of which is emergence: Definition (Emergence) For Butterfield and Isham, a theory T 1 emerges from another theory T 2 iff there exists either a limiting or an approximating procedure to relate the two theories (or a combination of the two).

34 Limiting and approximating procedures Definition (Limiting procedure) A limiting procedure is taking the mathematical limit of some physically relevant parameters, in general in a particular order, of the underlying theory in order to arrive at the emergent theory. won t work, at least not alone, because of technical problems (maximal loop density; Rovelli: more loops give more size, not a better approximation to a given [classical] geometry (2004, 6.7.1)), conceptual problems (superpositions don t vanish)

35 Definition (Approximating procedure) An approximating procedure designates the process of either neglecting some physical magnitudes, and justifying such neglect, or selecting a proper subset of states in the state space of the approximating theory, and justifying such selection, or both, in order to arrive at a theory whose values of physical quantities remain sufficiently close to those of the theory to be approximated. How does selection of subset of states happen? Is there a mechanism that would drive the system to the right states? Candidate mechanism: decoherence

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37 Applying the Butterfield-Isham scheme Thesis At least to the extent to which LQG is a consistent theory, (a close cousin of) GR can be seen to emerge from LQG if a delicately chosen ordered combination of approximations and limiting procedures is applied. Let me sketch how preliminary work by physicists might bear out this scheme...

38 Note: given the lack of understanding of dynamics, confine ourselves to kinematic states Rough idea for constructing semi-classical states: select kinematical states which correspond to almost flat three-metrics (three-geometries with small quantum fluctuations). Several approaches: 1 weave states (Ashtekar, Rovelli, Smolin 1992) 2 coherent states (Thiemann at el.) 3 photon Fock states (Varandarajan) 4 shadow states (Ashtekar et al.) etc

39 Weave states weave : appear to be smooth when seen from afar, discrete structure when examined more closely eigenstates of geometrical operators for volume of region R with eigenvalues that approximate corresponding classical volume of R also eigenstates of geometrical area operator for surface S Technically: consider macroscopic 3-dim region R with 2-dim surface S and 3-dim gravitational field e i a( x) defined for all x R This gravitational field defines metric field q ab ( x) = e i a( x)e j b ( x)η ij ( x) for which it is possible to construct a spin network state S such that S approximates the metric q ab for sufficiently large scales l Pl

40 Classical area and volume Classically, the area of a two-dimensional surface S M and the volume of a three-dimensional region R M with respect to a fiducial gravitational field 0 ea i are given by (Rovelli 2004, 2.1.4) A[ 0 e, S] = d 2 S, (6) V[ 0 e, R] = d 3 R, (7) where the relevant measures for the integrals are determined by the (typically flat) 0 e i a.

41 Quantum approximation: weave states The requirement that the spin network state S must approximate the classical geometry for sufficiently large scales is made precise by demanding that S be a simultaneous eigenstate of the area operator  and the volume operator V with eigenvalues equal to the classical values as given by (6) and (7), respectively, up to small corrections of the order of l Pl / : Â(S) S = ( A[ 0 e, S] + O(l 2 Pl / 2 ) ) S, (8) V(R) S = ( V[ 0 e, R] + O(l 3 Pl / 3 ) ) S. (9) At scales much smaller than, quantum features are relevant; at scales of order or larger, weave states exhibit a close approx to corresponding classical geometry in that it determines the same areas and volumes as the classical metric q ab weave states are semi-classical approximations (CSMC)

42 Limiting procedure involves taking the limit l Pl / 0 will make the small corrections in (8) and (9) disappear can be done either by letting (i.e. have size of R grow beyond limits), or l Pl 0 (corresponds to 0)

43 Decoherence reconsidered problem with decoherence as mechanism to drive selection of quantum states to weave states: states represent all of space, no environment conceive of area and volume operators as local properties of quantum gravitational field (just as local geometrical properties in GR) only consider medium-sized chunk of space(time) in our lab re-opens door for invoking decoherence as potential justification of selection: measurement interaction may force quantum state of measured piece of space into weave state MP must be addressed

44 Conclusion I have shown how classical space and time disappear in CQG, and how they might be seen to re-emerge from the fundamental, non-spatiotemporal structure. particularly: even though a case must be made for applicability of traditional measurement concept, the way in which classicality emerges from the quantum thy does not radically differ from ordinary QM Project relevant and interesting for at least two reasons: 1 important foundational questions concerning the interpretation of, and the relation between, theories are addressed, which can lead to conceptual clarification of the foundations of physics 2 relevant consequences for specifically philosophical (particularly metaphysical) issues are studied QG is fertile ground for the metaphysician